#143 X marks the Spot
(set on 11 December) Solution
The patio requires 196 white tiles and 29 black tiles. Here’s why:
Suppose the patio is a k × k square containing a total of k2 tiles. If k is odd we need 2k – 1 black tiles and so k2 – 2k + 1 = (k – 1)2 white tiles (a square number); and if k is even, (k – 1)2 – 1 (which is one less than a square). But no permutation of 1, 6 and 9 is one less than a square; so we are looking for a square for which k-1 is even. Of the three squares 169, 196 and 961 only 196 = 142 is even; so (k – 1)2 = 196 = 142 and k = 15. We need 2k – 1 = (2)(15) – 1 = 29 black tiles and 196 white tiles to make a 15 × 15 square.
#144 Shaken, not stirred
set by Alison Kiddle
On special occasions, Grace enjoys a perfect martini with 6 parts gin to 1 part vermouth. Unfortunately, she has been gifted two bottles of ready mixed martini, one in a ratio of 5:1 and one in a ratio of 7:1, neither of which is to her taste. However, she has a measuring jug that measures multiples of 100ml, and an empty bottle.
How much of each blend should she use to create her perfect martini?
SolutionThere is a trap to avoid here, as a 6:1 gin to vermouth ratio feels like it should be halfway between 5:1 and 7:1, but Grace tried mixing equal quantities and it tasted wrong.
The two bottles are 5/6 gin and 7/8 gin, respectively, and Grace wants a mixture that is 6/7 gin. That means 300 millilitres of the first bottle would have 250ml gin and 50ml vermouth, and 400ml of the second bottle would have 350ml gin and 50ml vermouth, which when mixed together gives 700ml of martini in Grace’s perfect 6:1 ratio. Cheers!
#145 Up the sprout
set by Rob Eastaway
My local greengrocer has a strange way of weighing produce. Vegetables go in a pan attached to a string that goes around a frictionless pulley and the pulley hangs from a suspended spring scale. The other end of the string is secured to the floor.
I picked out some sprouts for my Christmas dinner, at the bargain price of ÂŁ1 per kilogram. The weighing scale indicated 1.6kg.
How much do the sprouts weigh? And how much do I owe?
Solution
The sprouts weigh 0.8 kilograms so I owe 80p. The scales are experiencing 0.8kg weight from the sprouts and a balancing 0.8kg from the tension in the string that is attached to the floor.
If you don’t believe this answer, set up a pulley and a scale and test this out for yourself.
#146 Meg’s pegs
set by Colin Beveridge
Meg and Greg are playing a guessing game. Meg picks a code of four coloured pegs, each of which may be red, yellow, green or blue.
Greg’s first, incorrect, guess at Meg’s code is red-red-blue-green.
Meg tells Greg how many pegs are the correct colour in the correct place. She then tells him how many of the remaining pegs are a correct colour in the wrong place.
“Interesting!” laughs Greg. “In that case, I know your code.”
What is Meg’s code?
Solution
Meg’s code must be yellow-yellow-yellow-yellow. The only way that Greg can be certain of Meg’s code is if Meg says Greg had zero correct colours.
For any other combination of correct colours and positions, there are at least two possible codes.
#147 Lebkuchen race
set by Steve Wain
A German company sells its festive lebkuchen in tube-shaped packs of 15 biscuits. Made with mathematical perfection, the diameter of each biscuit is exactly five times its thickness.
As I unpacked my shopping, I noticed two ants on the top rim of the pack I’d just bought. They set off at the same moment on what looked like a race. One marched along the length of the packet while the other, moving at the same speed, began on a circuit around the top of the packet.
Which ant finished first?
Solution
The ant walking the length of the packet won the race, because its journey was slightly shorter than that of the ant walking around the circumference of the packet’s end.
If we call the diameter of a biscuit D, then the length of the packet is 15 x D/5 = 3D, while the circumference is πD = 3.14 D.